This is a response to Harvey Friedman's' "Intrinsic interest of
effective bounds" which, in turn, is a reply to my "Deep Thought
about number theory."
The quotes are from Harvey's message. The replies are mine.
> I conjecture that...your "Deep Thought" would agree with me if I
> talked to him/her, as the exact way in which these questions are
> phrased is extremely important;
I think you've done yourself in here. I have the same suspicion
about you. Even if Deep Thought were to agree with you, what would
that necessarily demonstrate other than your rhetorical skills?
> As a test, I would ask your "Deep Thought" whether a result to the
> effect that "there is no effective bound whatsoever" is intrinsically
interesting.
"No bound" is a very different matter, especially psychologically.
Remember the phenomenon of "Isn't it fascinating? We proved it exists
but can't find it."
> I believe that this is more than enough to justify the intrinsic
> interest of the negation - i.e., the existence of an effective
> bound.
Harvey, this is such an old move. But I grant that, logically,
there is something to it. It's like the paradox of confirmation. I
want to get evidence for the thesis that all swans are white. So I
look all over my house checking each non-white thing and finding,
"Yep, it's not a swan." (Except that, in your case, the checking is
usually a whole lot more tedious.)
> The natural progress of mathematics is to first prove something new,
> and then to build on that success to prove something better, and then
> to build on that success to prove something still better, etcetera.
This doesn't help your cause. Even if every good thing is of this
kind, it doesn't follow that everything of this kind good.
(For a trivial example, suppose the enterprise consists in getting
ever better approximations to the first sign change of pi - li by first
getting a "not desirable" one to within 2, then a "more desirable" one
to within 1.5, then an "even more desirable" one to within 1.25, etc..)
> The alternative would be to hold that unless one has a good bound from
> the beginning, one should not disseminate any not-so-good bound. That
> attitude goes counter to the standard mode of operation in mathematics,
> science, engineering, business, etcetera.
The alternative is that maybe there are better ways to spend 22
years than looking for a not-so-good bound, especially when there
aleady is one in the proof.
With best regards,
Gabriel